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\examtitle{MATH 2270-2}{Final Exam Sample Problems}{Spring 2012}

\studentinfo
\instructions{No books, notes or electronic devices, please.\\
The questions have credits which reflect the time required to write the solution.\\
If you must write a solution out of order or on the back side, then supply a road map.\\
Solutions are expected to include readable and convincing details. A correct answer without details earns 25\%.\\
Only a small sampling from these problem types will appear on the final exam, due to time limitations.
Expect about 10 minutes per problem. A final exam problem will have multiple parts. 
}
\finishfirstpage
% Problems Start Here % ----------------------------------------------------- %
\section*{Chapters 1 and 2}

\problem{5}{
Let $A$ be a $2\times 2$ matrix such that
$A\begin{pmatrix}1\\1\end{pmatrix}=\begin{pmatrix}1\\0\end{pmatrix}$. Compute
$A\begin{pmatrix}2\\2\end{pmatrix}$.
}{}{}

\anotherpage

\problem{5}{
State (1) the definition of norm, (2) the Cauchy-Schwartz inequality and (3) the triangle inequality, for vectors in $\Rr^n$.
}{}{}

\problem{5}{
Define an Elementary Matrix. Display the fundamental matrix multiply equation which summarizes a sequence of swap, combo, multiply operations, transforming a matrix $A$ into a matrix $B$.
}{}{}

\anotherpage

\problem{5}{
Suppose $A=B(C+D)E$ and all the matrices are invertible. Find an equation for $C$.
}{}{}

\anotherpage

\problem{5}{
Let $A=\twobytwo{2&1}{0&3}$. Show the details of two different methods for finding $A^{-1}$.
}{}{}

\anotherpage

\problem{5}{
Find a factorization $A=LU$ into lower and upper triangular matrices for the matrix $A=\threebythree{2&1&0}{1&2&1}{0&1&2}$.
}{}{}

\problem{5}{
Let $Q$ be a $2\times 2$ matrix with $QQ^T=I$. Prove that $Q$ has columns of unit length and its two columns are orthogonal.
}{}{}


\anotherpage

\section*{Chapters 3, 4}

\problem{5}{
Let $V$ be a vector space and $S$ a subset of $V$. State the {\bf Subspace Criterion}, a theorem with three requirements, and conclusion that $S$ is a subspace of $V$.
}{}{}

\problem{5}{
Explain how the {\bf span theorem} applies to show that the set $S$ of all linear combinations of
the functions $\cosh x, \sinh x$ is a subspace of the vector space $V$ of all continuous functions on $-\infty < x < \infty$.
}{}{}

\problem{5}{
Write a proof that the subset $S$ of all solutions $\vec{x}$ in $\Rr^n$ to a homogeneous matrix equation $A\vec{x}=\vec{0}$ is a subspace of $\Rr^n$. This is called the {\bf kernel theorem}.
}{}{}

\problem{5}{
Using the subspace criterion, write two hypotheses that imply that a set $S$ in a vector space $V$ is not a subspace of $V$. The full statement of three such hypotheses is called the {\bf Not a Subspace Theorem}.
}{}{}

\problem{5}{
Report which columns of $A$ are pivot columns: $A=\threebythree{0&1&1}{0&1&2}{0&0&0}$. 
}{}{}

\problem{5}{
Find the complete solution $\vec{x} = \vec{x}_h + \vec{x}_p$ for the nonhomogeneous system
$$\threebythree{0&1&1}{0&1&2}{0&0&0}\colvector{x_1 \\ x_2 \\ x_3} = \colvector{2 \\ 3 \\ 0}.$$
The homogeneous solution $\vec{x}_h$ is a linear combination of Strang's special solutions.
Symbol $\vec{x}_p$ denotes a particular solution.
}{}{}

\problem{5}{
Find the reduced row echelon form of the matrix $A=\threebythree{0&1&1}{0&0&0}{0&1&2}$.
}{}{}

\problem{5}{
A $10\times 13$ matrix $A$ is given and the homogeneous system $A\vec{x}=\vec{0}$ is transformed to
reduced row echelon form. There are $7$ lead variables. How many free variables? 
}{}{}

\problem{5}{
The rank of a $10\times 13$ matrix $A$ is $7$. Find the nullity of $A$.
}{}{}


\problem{5}{
Let $S$ be the subspace of $\mathbb{R}^4$ spanned by the vectors
$\bar{v}_1=\begin{pmatrix}1\\1\\1\\0\end{pmatrix}$ and
$\bar{v}_2=\begin{pmatrix}1\\1\\0\\1\end{pmatrix}$. Find the Gram-Schmidt orthonormal basis
of $S$.
}{}{}
\anotherpage

%\problem{5}{
%%%Not on the final exam S2012:
%Given a basis $\vec{v}_1=\begin{pmatrix}3\\2\end{pmatrix},
%\vec{v}_2=\begin{pmatrix}4\\4\end{pmatrix}$ of $\Rr^2$, and
%$\vec{v}=\begin{pmatrix}10\\4\end{pmatrix}$, then
%$\vec{v}=c_1\vec{v}_1+c_2\vec{v}_2$ for a unique set of coefficients
%$c_1,c_2$, called the {\em coordinates of $\vec{v}$ relative to the
%basis $\vec{v}_1, \vec{v}_2$}. Compute $c_1$ and $c_2$.
%}{}{}


\problem{5}{
Let the linear transformation $T$ from $\Rr^3$ to $\Rr^3$
be defined by its action on three independent vectors:
Given a basis $$
T\left(\begin{pmatrix}3\\ 2\\ 0\end{pmatrix}\right)=
\begin{pmatrix}4 \\ 4\\ 2\end{pmatrix}, 
T\left(\begin{pmatrix}0\\ 2\\ 1\end{pmatrix}\right)=
\begin{pmatrix}4 \\ 0\\ 2\end{pmatrix},
T\left(\begin{pmatrix}1\\ 2\\ 1\end{pmatrix}\right)=
\begin{pmatrix}5 \\ 1\\ 1\end{pmatrix}.$$
Find the unique $3\times 3$ matrix $A$ such that $T$ is defined
by the matrix multiply equation $T(\vec{x})=A\vec{x}$.  
}{}{}

\anotherpage
\problem{5}{
 Determine independence or dependence for the list of vectors \[\begin{pmatrix} 1 \\ 2 \\ 3\end{pmatrix}, \begin{pmatrix} 4\\ 0\\ 4\end{pmatrix}, \begin{pmatrix} 3 \\ 2
  \\1\end{pmatrix}\]
}{}{}

\anotherpage


\problem{5}{
Use least squares to find the best fit line for the
points $(1,2)$, $(2, 2)$, $(3,0)$.
}{}{}
\anotherpage

\problem{5}{
Find all solutions to the system of equations
\begin{align*}
  2w+3x+4y+5z = 1\\
 4w+3x+8y+5z = 2\\
 6w+3x+8y+5z = 1\\
\end{align*}
}{}{}

\anotherpage

\problem{5}{
The spectral theorem says that a symmetric matrix $A$ can be factored into $A=QDQ^T$ where $Q$ is orthogonal and $D$ is diagonal. Find $Q$ and $D$ for the symmetric matrix
$A=\begin{pmatrix}3 & -1\\ -1 &3\end{pmatrix}$.
}{}{}

\anotherpage

\problem{5}{
Show that if $B$ is an invertible matrix and $A$ is similar to $B$, with $A=PBP^{-1}$,
then $A$ is invertible.
}{}{}

\anotherpage

\problem{5}{
Prove that the null space $S$ of an $m\times n$ matrix $M$ is a subspace
of $\mathbb{R}^n$. This is called the {\em Kernel Theorem}, and it is
proved from the {\em Subspace Criterion}.
}{}{}

\anotherpage
\problem{5}{
Let $A$ be an $m\times n$ matrix with independent columns. Prove that
$A^TA$ is invertible.
}{}{}

\anotherpage
\problem{5}{
Let $A$ be an $m\times n$ matrix with $A^TA$ invertible. Prove that the
columns of $A$ are independent.
}{}{}

\anotherpage
\problem{5}{
Let $A$ be an $m\times n$ matrix. Denote by $S_1$ the row space of of $A$ and $S_2$ the column space of $A$.
Prove that $T ~:~ S_1 \to S_2$ defined by $T(\vec{x})=A\vec{x}$ is one-to-one and onto.
}{}{}

\anotherpage
\problem{5}{
Let $A$ be an $m\times n$ matrix and $\vec{v}$ a vector orthogonal to the nullspace of $A$. Prove that $\vec{v}$ must be in the row space of $A$.
}{}{}


\anotherpage
\problem{5}{
State the Fundamental Theorem of Linear Algebra. Include {\bf Part 1}: The dimensions of the four subspaces, and {\bf Part 2}: The orthogonality equations for the four subspaces.
}{}{}

\anotherpage
\problem{5}{
Display the equation for the Singular Value Decomposition (SVD), then
cite the conditions for each matrix.
}{}{}

\anotherpage
\problem{5}{
Display the equation for the pseudo inverse of $A$, then define and
document each matrix in the product.
}{}{}

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